Researching Lesson Study with The Anthropological Theory ...

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Institut for Naturfagenes Didaktik

APEC CONFERENCE FEB. 2017, U. OF TSUKUBA, TOKYO

Researching Lesson Study withThe Anthropological Theory of the Didactic

Carl Winsløw, [email protected]

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1. LS as a ”research topic”à Need for scientific models of LS

2. Model of ”what LS is about” à mathematical and didactic praxeologies

3. Model of ”what LS is” à institutions and infrastructure

4. Some results of ATD research on LS5. The Anthropological Theory of the Didactic (ATD)à Outlook and invitation

6. Conclusion

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149 research publications in mathematics education journals and books with “Lesson Study” in title

1 – Lesson Study as Research topic

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Some overall research directions pursued:1 – Lesson Study as Research topic

Examplesof work

OBJECT:授業研究 in Japan

OBJECT:Lesson study (…) abroad

Descriptiveresearch: what is lesson study?

Lewis (1998): A lesson is like a swiftly flowing riverIsoda, Stephens, Ohara, Miyakawa (2007): Japanese Lesson Study in Mathematics

Yoshida (2012): Mathematics lesson study in the United States: Current status and ideas…Hart, Alston, Murata (2011): Lesson Study Research and Practice in Mathematics Education

Interven-tionresearch:Results of specific LS,LS as method

授業研究 itself

Rasmussen (2016): Lesson study in prospective mathematics teacher education: didactic and paradidactic technology in the post-lesson reflection.

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Descriptive research considers LS as an object of study with research questions like- what does LS consist of (description of « interior »)- what are the essential parts (« boundary »)- what are the essential conditions for LS (« external »)- what is LS about (what are its objects and aims)

Intervention research uses LS as a method to studymore specific questions about mathematics teachingand learning, teacher education etc.

In both cases, researchers need explicit models of LS and its « surroundings » in order to make theirquestions, methods and results precise, intelligible and open to scientific debate (including critique).

1 – Lesson Study as Research topic

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Scientific Models

All models are wrong but some are useful(George E.P. Box)

”complex reality”

Model: ”simple” structure, described within a theory(i.e. a scientific register)

2 – Model of what LS is about

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What is LS about?

FIRST NAIVE ANSWERS

Object of LS (what it is ”about”)- Teaching, by teachers (more specifically, a lesson)- Learning by studentsObjective of LS : to promote- Learning, by teachers, related to the previous two objects.

LS thus is about very specific forms of human practice and knowledge:- Mathematical Practice and Knowledge (by students, teachers)- Didactic Practice and Knowedge (by teachers)

Here, ”didactic” refers to the act of teaching or, more broadly, ”inducing others into a certain practice / knowledge”. ”Didactic practice” is fundamental to all human practices!


PRAXEO

LOG

Y

The notion of praxeology (praxis + logos)

DidMatV

Theory

(Chevallard, 1999)

PRACTICAL-BLOCK: ‘know how’Tasks, solved by techniques

THEORETICAL BLOCK: ‘know that and why’Techniques can bedescribed and justified

Technology (techne + logos)

Technique

Type of tasks

Example of a mathematical praxeology:- Type of task: find all x satisfying �� (for given �, �, �)- Technique: computation based on � � ��

��- Technology: explanation of formula (how, when, why,…))- Theory: algebraic definitions, rules, etc. to justify

technology


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Mathematical and Didactic Praxeologies

Mathematical Praxeology (MP): this is what the students should learn; we can model it with the precision we need for a given research project

Didactic Praxeology (DP): 1. the didactic practice (tasks and techniques) can be observed:

- tasks of teaching (related to some MP)- techniques (what the teacher does to some those tasks)

2. the corresponding didactic technology and theory is not observable in the classroom – the teacher (normally) neitherexplains nor justifies his practice there.

NOTICE: - A DP is always intimately related to a MP (for math. teaching) - technology and theory of DP ~ core of teacher knowledge- The tendency of didactic technology and theory to be

”personal”, ”non-shared”, fragmented etc.


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What is LS about? Praxeological models


Object of LS (what it is ”about”)- Didactic Practice (planned, observed, discussed)- Mathematical Praxeologies (of students – in LS,

planned/anticipated, observed and discussed)Objective of LS : to develop teachers’ didacticknowledge (technology, theory)- Shared didactic technology and theory needed,

and developed, in both planning and reflectionmeetings

An example of analysing a lesson study activity(mainly lesson plan, lesson and reflection) can befound in Miyakawa & Winsløw, JMTE 2013.

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3 – Model of what LS isLS is often described in terms of ”cycles” or otherindications of a process that unfolds in time

But to analyse our observations of a specific LS, this model is far too simple. We are interested in the specific MP and DP at stake. So what is LS itself, relatively to these?

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Mathematical infrastructure (for MP)Works as ”facility” for mathematical practiceEx1 ℂ used to compute rotation by αEx2 Computer algebra systemDidactic infrastructure (for DP)Artefacts and ressource systems as facility for didacticpracticeExs: textbook, smartboard, CAS, lesson plan,…

All human activity suppose what I call a praxeologicalinfrastructure… [which] comprises in particular somesmall and large facilities [dispositifs] which are works, and which allow the development of superstructuralactivities – the execution of some technique beingsupported by an infrastructure… (Chevallard 2009, summer school at Clermont-Ferrand)

3 – Model of what LS is

Didactic practiceParadidactic practice

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Teachers’ work in classroom

Teachers’ work outsideclassrooom

Paradidactic infrastructure: in addition to some ofthe didactic infrastructure, special ”facilities” for thework of t, such as office space, teacher guide,personal notes etc. Paradidactic praxeologiesdevelop personal knowledge of t.How about institutionalization?


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Elements of paradidactic infrastructure in Japan:• Konaikenshyuu (study groups of teachers within

each school), with grants, activity reports etc.• Lesson study• Mathematical circles (teacher seminar accross

schools)• Open lessons (municipal, regional national) with

invited and registered participants• Conferences with invited and registered

participants• Teacher journals and books• School math dictionnary• Videos with ”super-lessons”• TV-shows with famous teachers

(”fun math” with kids)


time

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PARADIDACTICINFRASTRUCTURE

LESSON

Post-didacticpraxeology[discussion]

Teachers analyse

observed DP,MP

MP developed in class

DPdeveloped in class

Paradidactic infrastructure of a school institution: the total of conditions for teachers work outside ”class” (MW,2013)

Pre-didacticpraxeologyLesson plan

Teachers prepare lessonrelated to MP

OBSERVATION

The case of lesson study (Miyakawa & W, 2013)3 – Model of what LS is

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4 – Some results of ATD research on LS

Miyakawa & Winsløw (2013). For a specific LS, coherent analysis of “what teachers learn” (DP theory block), the lesson itself (DP practice), and “what students learn” (MP).

Rasmussen (PhD-thesis, 2015). Analysis of postdidacticpraxeology in LS embedded in multidisciplinary teacher education, showing the potential for developing didactic knowledge “to the benefit of prospective teachers, educators and researchers alike”.

Østergaard (PhD-thesis, 2016). Analysis of the common “gap” between DP practice and knowledge blocks, and between MP and DP, as developed in teacher education … interventions with LS has strong potential to bridge those gaps

Carlsen (PhD-thesis, forthcoming). During practicum LS, teacher students develop (new) didactic technology about CAS-use in lower secondary algebra.

Bahn (Phd-thesis, forthcoming). Experimenting, with success, a concrete paradidactic infrastructure for implementing LS in DK.

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An example (Miyakawa & Winsløw, 2013)School festival, June 2009

• Primary school attached to Joetsu U.• Held every year at this school (Friday+Saturday)• 70 open lessons in all subjects, in two days• 700 guests in 2009 (from all of Japan) – teachers and…• All aspects of the school life on display, but the primary

aim is to share and develop knowledge for teaching


The lesson• Grade 2 (students about 7 years old), 40 students• Lesson no. 16 in a series of 17 lessons, duration 65 mins• Series title: Sukkiri as we draw!

(sukkiri ≈ experience of clarity, « aha »)• Japanese « open approach » (Nohda, 2004)• Student assignment : find solution and make a drawing

which, to a group of students, give « clarity » (of solution and method, and on who are convinced of what and why)

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A mathematical task, wide choice of techniques

Task• [t0 : to formulate t] [didactic technique]• t (the task) : There were 16 persons on a bus. Later,

more people got on, so there were 34 in total. How manyhad got on the bus ?

• T (type of task) : Given a total N and a part A, find rest B (N = A + B; given N and A, find B).

Techniques• τ0 : identify numbers (16, 34); choose the right operation

(+/-); execute (results: 16+34 or 34-16 or …)• τ1 : represent situation symbolically and solve

(34 = 16 + rest à rest = 34 – 16).• τ2 : represent situation by a drawing (abstraction from

concrete situation) showing situation and solution « in one » (as opposed to arithmetic/algebraic approaches).

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Reflection meeting : the meaning of « sukkiri »

Main questions• Why this acitivity « sukkiri by drawing » ?• What is the meaning of « sukkiri » ?

Comment of a participant• « sukkiri » it’s to have a drawing that helps them to get

the answer, to set up a calculation, or to see what is the object of the interrogation sign – not to have drawingwhich depicts the problem exactly

The paradidactic practice• The « new » notion of sukkiri (a priori not didactic

technology) facilitates an shared development of didactictechnology and theory, focusing on the production and use (by students) of more or less non-standard mathematical ostensives

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Reflection meeting : techniques for managing the students’ presentations

Participants• Evoke pupil mathematical techniques observed in

the lesson (e.g., 34 + 16) and possible explanations (e.g., the wording of the problemimplies addition since « the whole » ~ sum) ;

• Propose alternative or motified didactic techniques for an DO in which the task is to teach « drawingtechniques » for subtraction (τ2).

The paradictic practice• Question the planned and observed DO by taking

into accound the MO realised in the class, as wellas the didactic technology of the lesson plan

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Reflection: theory levelPrincipal question

• To what extent are the choises and arguments of the planned DP justified?

Justifications mentioned• National programme (curriculum);• Philosophy of this school (« prepare the pupils to live in

human society ») ; à Comment of the guide (prof. Nunokawa): the

importance of interaction in society, even if you have a strong personality (« sukkiri »)

Paradidactic practice• The discussion allows for reflection on more general

didactic principles, firmly based on the commonobservation of a concrete DP

• Development of possible new didactic technology(”sukkiri”)

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5 – The anthropological theory of the didactic

ATD welcomes you with:- An elaborate system of theoretical

tools to model and design mathematical and didactical knowledge and practice

- A research community with active researchers in many countries (Japan, Korea, France, Spain, Germany, Denmark, Sweden, Brazil, Canada, USA, …), international congresses etc.

- A research programme with a tight network of questions, methods, results and so on

- A definite potential for LS research

Conclusion: the profession of teachingEtzioni, quoted by Chevallard (2009):

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6 – Conclusion

Crucial: existence of paradidactic infrastructure!… to enable a collective development of knowledge(DP technology and theory), as in LS.

Thanksありがとう


Researching Lesson Study with The Anthropological Theory ...

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